The Database Migration Service is typically used for moving databases into the cloud, not for ongoing connection management. While a self-hosted proxy on a compute instance could control connections, it can lead to higher operating costs and increased administrative effort. PrivateLink focuses on secure, private connectivity and does not offer features for efficient connection handling such as pooling. RDS Proxy, however, is specifically implemented for managing large numbers of concurrent connections, significantly reducing the database workload and scaling challenges. It effectively helps in maintaining low latency and cost-efficiency by pooling and sharing established connections without the overhead of establishing new ones each time.